Poisson equation in domains with concentrated holes

Abstract

We consider solutions $u^\varepsilon$ of Poisson problems with the Dirichlet condition on domains $\Omega_\varepsilon$ with holes concentrated at subsets of a domain $\Omega$ non-periodically. We show $u^\varepsilon$ converges to a solution of a Poisson problem with a simple function potential. This is a generalized result of a sample model given by Cioranescu and Murat (1997). They showed a result for case that holes are distributed at $\Omega$ periodically.